Semisimple Lie ideal has a commuting ideal complement ($L⊴M⟹M=L\oplus L^{{\perp }_{{\kappa }_M}}$)

A finite-dimensional complex semisimple Lie algebra which is an ideal in a finite-dimensional Lie algebra has the displayed direct complement. The restriction of the Killing form of $M$ to $L$ equals the Killing form of $L$: the induced adjoint maps of elements of $L$ on $M/L$ vanish. This restriction is nondegenerate by the Cartan criterion for semisimplicity. Invariance makes its orthogonal complement an ideal. Nondegeneracy on $L$ gives a direct vector-space sum with that complement, and the bracket of the two ideals lies in their zero intersection, so they commute. The ambient algebra need not itself be semisimple.